Capital Structure Arbitrage: Model Choice and Volatility Calibration

44 Pages·2007·0.54 MB·English

Capital Structure Arbitrage: Model Choice and
Volatility Calibration
(cid:3)
Claus Bajlum Peter Tind Larsen
y z
First Draft: August 28, 2006
Current Version: January 10, 2007
Abstract
When identifying relative value opportunities across credit and equity
markets, the arbitrageur faces two major problems, namely positions based
on model misspeci(cid:133)cation and mismeasured inputs. Using credit default
swap data, this paper addresses both concerns in a convergence-type trad-
ing strategy. In spite of di⁄erences in assumptions governing default and
calibration, we (cid:133)nd the exact structural model linking the markets second
to timely key inputs. Studying an equally-weighted portfolio of all rela-
tive value positions, the excess returns are insigni(cid:133)cant when based on a
traditional volatility from historical equity returns. However, relying on an
impliedvolatilityfromequityoptionsresultsinasubstantialgaininstrategy
execution and highly signi(cid:133)cant excess returns - even when small gaps are
exploited. The gain is largest in the speculative grade segment, and cannot
be explained from systematic market risk factors. However, although the
strategy may seem attractive at an aggregate level, positions on individual
obligors can be very risky.
We thank Lombard Risk for access to the credit default swap data. We are grateful to
(cid:3)
Peter Lłchte Jłrgensen, Hayne Leland, Stephen Schaefer, Ilya Strebulaev, Carsten Słrensen,
participants at the C.R.E.D.I.T. 2006 Doctoral Tutorial in Venice, the Danish Doctoral School
of Finance Workshop 2007 and seminar participants at Aarhus School of Business for useful
discussions and comments. Any remaining errors are our own.
DanmarksNationalbankandCopenhagenBusinessSchool,DepartmentofFinance,Solbjerg
y
Plads, 2000 Frederiksberg, Denmark. E-mail: cb.(cid:133)@cbs.dk, tel.: [phone].
School of Economics and Management, University of Aarhus, Universitetsparken 322, 8000
z
Aarhus C, Denmark. E-mail: [email protected], tel.: [phone].
1
1 Introduction
Capital structure arbitrage refers to trading strategies that take advantage of the
relative mispricing across di⁄erent security classes traded on the same capital
structure. As the exponential growth in the credit default swap (CDS) market
has made credit much more tradable and traditional hedge fund strategies have
su⁄ered declining returns (Skorecki (2004)), important questions arise for hedge
funds and proprietary trading desks. In particular, do credit and equity markets
ever diverge in opinion on the quality of an obligor? What is the risk and return of
exploiting divergent views in relative value strategies? Although trading strategies
founded in a lack of synchronicity between equity and credit markets have gained
huge popularity in recent years (Currie & Morris (2002) and Zuckerman (2005)),
the academic literature addressing capital structure arbitrage is very sparse.
This paper conducts a comprehensive analysis of the risk and return of capi-
tal structure arbitrage using CDS data on 221 North American obligors in 2002
to 2004. When looking at one security in order to signal the sale or purchase of
another, the resulting link and initiation of a trade depends on the chosen model.
We address two major problems facing the arbitrageur, namely relative value op-
portunities driven by model misspeci(cid:133)cation or mismeasured inputs.
Duarte,Longsta⁄&Yu(2005)analyzetraditional(cid:133)xedincomearbitragestrate-
gies such as the swap spread arbitrage, but also brie(cid:135)y address capital structure
arbitrage. Yu (2006) argues for a complete lack of evidence in favor of or against
strategies trading equity instruments against CDSs. He conducts the (cid:133)rst analy-
sis of the strategy by implementing the industry benchmark CreditGrades with a
historical volatility, as reputed used by most professionals.
We show that the more comprehensive model by Leland & Toft (1996) only
adds an excess return of secondary order. However, when exploiting a wider array
of inputs and securities in model calibration and identi(cid:133)cation of relative value
opportunities, the result is a substantial improvement in strategy execution and
returns.
When searching for relative value opportunities, the arbitrageur uses a struc-
tural model to gauge the richness and cheapness of the 5-year CDS spread. Using
the market value of equity, a volatility measure and the liability structure of the
obligor, he compares the market spread with the spread implied from the model.
When the market spread is substantially larger(smaller) than the theoretical coun-
terpart, he sells(buys) a CDS and sells(buys) equity. If the market and model
spreads subsequently converge he pro(cid:133)ts. Hence, the model helps identify credits
1
that either o⁄er a discount against equities or trade at a very high level.
As pointed out in Duarte et al. (2005), the nature of capital structure arbitrage
puts a premium on models that can explain the link between securities with dif-
ferent characteristics. In fact, the chosen underlying model plays a central role in
all parts of the strategy. First, it is used to calculate predicted CDS spreads gov-
erning entry and exit decisions in markets. Second, to calculate daily returns on
an open position, it is necessary to keep track on the total value of an outstanding
CDS position. This is done from the model-based term structure of survival prob-
abilities. Third, the model is used to calculate the equity hedge by a numerical
di⁄erentiation of the value of the CDS position wrt. the equity price.
CreditGrades loosely builds on Black & Cox (1976), with default de(cid:133)ned as
the (cid:133)rst passage time of (cid:133)rm assets to an unobserved default barrier. This model,
like other structural models, is based on a set of restrictive assumptions regarding
the default mechanism and capital structure characteristics.
Although allowing for a random recovery, CreditGrades belongs to the class of
models with an exogenous default barrier. However, Leland (1994) subsequently
extended in Leland & Toft (1996) has pioneered models with endogenous default.
In these models, the default barrier is chosen by managers as the asset value where
it is no longer optimal for the equityholders to meet the promised debt service
payments. Hence, the default barrier is determined not only by debt principal,
but also by asset volatility, debt maturity, payout rates and tax rates etc.
As a result of model variations, di⁄erences in model calibration exist. For
structural models, this is particularly relevant as many key inputs are di¢ cult
to measure. Bypassing strict de(cid:133)nitions CreditGrades is developed for immedi-
ate application, while the calibration of Leland & Toft (1996) is more extensive.
Hence, the number and characteristics of parameters to be estimated, as well as
the method to infer the underlying asset value process and default barrier, di⁄er
across models.
Duarte et al. (2005) and Yu (2006) solely rely on CreditGrades calibrated
with a 1000-day historical volatility.1 When based on a large divergence between
markets and smaller gaps are ignored, both (cid:133)nd that capital structure arbitrage
is pro(cid:133)table on average. At the aggregate level, the strategy appears to o⁄er
attractive Sharpe ratios and a positive average return with positive skewness. Yet,
individual positions can be very risky and most losses occur when the arbitrageur
1That CreditGrades is the preferred framework among professionals is argued in Currie &
Morris (2002) and Yu (2006), while the CreditGrades Technical Document by Finger (2002)
advocates for the 1000-day historical volatility.
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shorts CDSs but subsequently (cid:133)nd the market spread rapidly increasing and the
equity hedge ine⁄ective.
Duetothesubstantialdi⁄erencesinmodelassumptionsandcalibration,thekey
observedgapbetweenthemarketandmodel spreadfuelingthearbitrageurmaybe
driven by model misspeci(cid:133)cation. Furthermore, key inputs may be mismeasured
sending the arbitrageur a false signal of relative mispricing. Hence, there is an
urgent need to understand how the risk and return vary with model choice and
calibration. These caveats are unexplored in Duarte et al. (2005) and Yu (2006).
We address these two major problems facing the arbitrageur, and study how
the characteristics of capital structure arbitrage vary with model choice and asset
volatility calibration. For this purpose, we apply the CreditGrades model and
Leland & Toft (1996). As the volatility measure is a key input to the pricing of
credit, weidentifyrelativevalueopportunitiesfromatraditional250-dayhistorical
volatility used extensively in the bond pricing literature, and a volatility measure
implied from equity options.
Based on anecdotal evidence using CreditGrades, Finger & Stamicar (2005a)
andFinger&Stamicar(2005b)showhowmodelspreadsbasedonhistoricalvolatil-
ities lag the market when spreads increase, while overpredicting the market as
spreads recover. However, the more responsive implied volatility substantially
improves the pricing performance. Cao, Yu & Zhong (2006) conduct a more com-
prehensive study using regression analysis and the CreditGrades model. They (cid:133)nd
that the more timely option-implied volatility dominates the historical measure
in explaining CDS spreads, and that the gain is concentrated among (cid:133)rms with
lowercreditratings. Whileanalyzingthedeterminantsof CDSspreadsandpricing
errors in CreditGrades, they are silent on the risk and return of capital structure
arbitrage.
As the arbitrageur feeds on large variations in credit and equity markets, these
insights suggest the implied volatility to lead to superior entry and exit decisions
and trading returns. Furthermore, the gain from a more timely credit signal is
expected to be largest for the obligors of most interest to the arbitrageur, namely
those in the speculative grade segment.
Hence, we implement the strategy on 221 North American industrial obligors
in 2002 to 2004. Case studies illustrate that while model choice certainly mat-
ters in identifying relative value opportunities, the volatility input is of primary
importance. The historical volatility may severely lag the market, sending the
arbitrageur a false signal of relatively cheap protection in the aftermath of a cri-
sis. The result is large losses for the arbitrager as market spreads continue to
3
tighten. Indeed, the implied volatility may result in the exact opposite positions,
with obvious consequences for the arbitrageur.
However, irrespective of model choice and volatility calibration, the strategy is
very risky at the level of individual obligors. Convergence may never happen and
the equity hedge may be ine⁄ective. This may force the arbitrageur to liquidate
positions early and su⁄er large losses.
When studying the risk and return at an aggregate level, we focus on holding
period returns and a capital structure arbitrage index of monthly excess returns.
Indeed, both models generally result in insigni(cid:133)cant excess returns, when cali-
brated with a traditional volatility from historical equity returns. However, the
gain from identifying relative value opportunities from option-implied volatilities
is substantial.
In a variant of the strategy based on CreditGrades, the mean holding period
return for speculative grade obligors increases from 2.64 percent to 4.61 percent
when implemented with option-implied volatilities. The similar numbers based
on Leland & Toft (1996) are 3.14 versus 5.47 percent. However, the incremental
return is much smaller for investment grade obligors.
Additionally,thecorrespondingexcessreturnsarehighlysigni(cid:133)cantwhenoption-
implied volatilities are used to identify opportunities - even when small gaps are
exploited. BasedonCreditGrades,themeanexcessreturnis0.44percentoninvest-
ment grade and 1.33 percent on speculative grade obligors, both highly signi(cid:133)cant.
The similar numbers when Leland & Toft (1996) is used to identify relative value
opportunities are 0.27 and 2.39 percent, both highly signi(cid:133)cant. Finally, we do
not (cid:133)nd the excess returns to represent compensation for exposure to systematic
market factors.
We conclude that while model choice does matter for the arbitrageur, is seems
secondtoproperlymeasuredkeyinputsinthecalibration. Hence,ifthearbitrageur
reliesonthedynamicsofoptionpriceswhenidentifyingrelativevalueopportunities
across equity and credit markets, the result is a substantial aggregate gain in
trading returns above the benchmark application of capital structure arbitrage in
Duarte et al. (2005) and Yu (2006).
This paper is based on the premise that structural models price CDSs reason-
ably well. Ericsson, Reneby & Wang (2006) (cid:133)nd that Leland (1994), Leland &
Toft (1996) and Fan & Sundaresan (2000) underestimate bond spreads consistent
with previous studies, but perform much better in predicting CDS spreads. In
fact, the resulting residual CDS spreads are uncorrelated with default proxies as
well as non-default proxies. Since the rationale for the strategy is to exploit a
4
lack of integration between various markets, the paper is also related to studies
on the lead-lag relationship among bond, equity and CDS markets like Hull, Pre-
descu & White (2004), Norden & Weber (2004), Longsta⁄, Mithal & Neis (2005)
and Blanco, Brennan & Marsh (2005). While the CDS is found to lead the bond
market, no de(cid:133)nitive lead-lagrelationshipexists betweenequityandCDSmarkets.
Additionally, Hogan, Jarrow, Teo & Warachka (2004) study statistical arbi-
trages, while Mitchell & Pulvino (2001) and Mitchell, Pulvino & Sta⁄ord (2002)
are important studies on merger and equity arbitrage. Finally, the paper is related
to Schaefer & Strebulaev (2004), who show that structural models produce hedge
ratios of equity to debt that cannot be rejected in empirical tests.
The remainder of this paper is organized as follows. Section 2 outlines the
trading strategy, while the data is presented in section 3. Section 4 presents the
underlying models and calibration, and section 5 illustrates some case studies.
Section 6 presents the aggregate results of the strategy, and section 7 concludes.
2 Trading Strategy
This section describes the trading strategy underlying capital structure arbitrage.
The implementation closely follows Yu (2006), to whom we refer for a more elab-
orate description. Since a time-series of predicted CDS spreads forms the basis of
the strategy, we start with a short description of how to price a CDS.
2.1 CDS Pricing
A CDS is an insurance contract against credit events such as the default on a
corporate bond (the reference obligation) by a speci(cid:133)c issuer (reference entity).
In case of a credit event, the seller of insurance is obligated to buy the reference
obligation from the protection buyer at par.2 For this protection, the buyer pays a
periodic premium to the protection seller until the maturity of the contract or the
credit event, whichever comes (cid:133)rst. There is no requirement that the protection
buyer actually owns the reference obligation, in which case the CDS is used more
for speculation rather than protection. Since the accrued premium must also be
2In practice, there may be cash settlement or physical settlement, as well as a possible
cheapest-to-deliver option embedded in the spread. However, we refrain from this complica-
tion. Credit events can include bankruptcy, failure to pay or restructuring.
5
paid if a credit event occurs between two payment dates, the payments (cid:133)t nicely
into a continuous-time framework.
First, the present value of the premium payments can be calculated as
T s
EQ c exp r du 1 ds , (1)
u (cid:28)>s
(cid:18) Z0 (cid:18)(cid:0)Z0 (cid:19) f g (cid:19)
where c denotes the annual premium known as the CDS spread, T the maturity
of the contract, r the risk-free interest rate, and (cid:28) the default time of the obligor.
EQ denotes the expectation under the risk-neutral pricing measure. Assuming
independence between the default time and the risk-free interest rate, this can be
written as
T
c P(0;s)q (s)ds, (2)
0
Z0
where P(0;s) is the price of a default-free zero-coupon bond with maturity s, and
q (s) is the risk-neutral survival probability of the obligor, P((cid:28) > s), at t = 03.
0
Second, the present value of the credit protection is equal to
(cid:28)
EQ (1 R)exp r du 1 , (3)
u (cid:28)<T
(cid:18) (cid:0) (cid:18)(cid:0)Z0 (cid:19) f g(cid:19)
where R is the recovery of bond market value measured as a percentage of par
in the event of default. Maintaining the assumption of independence between the
default time and the risk-free interest rate and assuming a constant R, this can be
written as
T
(1 R) P(0;s)q (s)ds, (4)
(cid:0) (cid:0) 00
Z0
where q (t) = dq (t)=dt is the probability density function of the default time.
(cid:0) 00 (cid:0) 0
The CDS spread is determined such that the value of the credit default swap is
zero at initiation
T T
0 = c P(0;s)q (s)ds+(1 R) P(0;s)q (s)ds, (5)
0 (cid:0) 00
Z0 Z0
and hence
T
(1 R) P(0;s)q (s)ds
c(0;T) = (cid:0) 0 00 . (6)
(cid:0) T P(0;s)q (s)ds
0 R 0
3Later,wefocusonconstantinterestratesR. Thisassumption,togetherwithindependencebe-
tweenthedefaulttimeandtherisk-freeinterestrate,allowsustoconcentrateontherelationship
between the equity price and the CDS spread. This is exactly the relationship exploited in the
relative value strategy.
6
The preceding is the CDS spread on a newly minted contract. To calculate
daily returns to the arbitrageur on open trades, the relevant issue is the value of
the contract as market conditions change and the contract is subsequently held.
To someone who holds a long position from time 0 to t, this is equal to
T
(cid:25)(t;T) = (c(t;T) c(0;T)) P(0;s)q (s)ds, (7)
t
(cid:0)
Zt
where c(t;T) is the CDS spread on a contract initiated at t with maturity date T.
Equation (7) can be interpreted as a survival-contingent annuity maturing at date
T, which depends on the model-speci(cid:133)c term structure of survival probabilities
q (s) through s at time t.
t
Finally, we follow Yu (2006) in de(cid:133)ning the hedge ratio (cid:14)(t;T) as
@(cid:25)(t;T)
(cid:14)(t;T) = N , (8)
(cid:3) @S
t
where S denotes the market value of equity at time t and N is the number of
t
shares outstanding.4 The choice of underlying model-framework and calibration is
discussed in section 4.
2.2 Implementation of the Strategy
Next, we brie(cid:135)y describe the trading strategy as implemented in Duarte et al.
(2005) and Yu (2006). Using the market value of equity, a volatility measure
and the liability structure of the obligor, the arbitrageur uses a structural model
to gauge the richness and cheapness of the CDS spread. Comparing the daily
spreads in the market with the theoretical spreads implied from the model, the
model helps identify credits that either o⁄er a discount against equities or trade
at a very high level.
If the market spread at a point in time has grown substantially larger than
the model spread (or vice versa), the arbitrageur sees an opportunity. It might be
that the credit market is gripped by fear and the equity market is more objective.
Alternatively, he might think that the equity market is slow to react and the CDS
spread is priced fairly. If the (cid:133)rst view is correct, he should sell protection and
4This de(cid:133)nition deviates slightly from the one in Yu (2006), since we formulate all models on
atotalvaluebasisandnotpershare. Equation(8)followsfromasimpleapplicationofthechain
rule.
7
if the second view is correct, he should sell equity. Either way, the arbitrageur
is counting on the normal relationship between the two markets to return. He
therefore takes on both short positions and pro(cid:133)ts if the spreads converge. This
relative value strategy is supposed to be less risky than a naked position in either
market, but is of course far from a textbook de(cid:133)nition of arbitrage.
Two important caveats to the strategy are positions initiated based on model
misspeci(cid:133)cation or mismeasured inputs. Such potential false signals of relative
mispricing are exactly what this paper addresses.
We conduct a simulated trading exercise based on this idea across all obligors.
Letting (cid:11) be the trading trigger, c the market spread and c the equity-implied
t 0t
model spread, we initiate a trade each day, if one of the following conditions are
satis(cid:133)ed
c > (1+(cid:11))c or c > (1+(cid:11))c : (9)
t 0t 0t t
Inthe(cid:133)rstcase, aCDSwithanotionalof$1andsharesworth$ (cid:14) 1areshorted.5
(cid:0) (cid:3)
In the second case, the arbitrageur buys a CDS with a notional of $1 and buys
shares worth $ (cid:14) 1 as a hedge.
(cid:0) (cid:3)
Since Yu (2006) (cid:133)nds his results insensitive to daily rebalancing of the equity
position, we follow his base case and adopt a static hedging scheme. The hedge
ratio is therefore set to correspond to the model CDS spread c when entering the
0t
position, and (cid:133)xed throughout the trade.
Knowing when to enter positions, the arbitrageur must also decide when to
liquidate. We assume that exit occurs when the spreads converge, de(cid:133)ned as
c = c , or by the end of a pre-speci(cid:133)ed holding period, which ever comes (cid:133)rst. In
t 0t
principle, the obligor can also default or be acquired by another company during
the holding period. Yu (2006) notes that in most cases the CDS market will
re(cid:135)ect these events long before the actual occurrences, and the arbitrageur will
have ample time to make exit decisions.6 Speci(cid:133)cally, it is reasonable to assume
that the arbitrageur will be forced to close his positions once the liquidity dries
up in the underlying obligor. Such incidents are bound to impose losses on the
arbitrageur.
5(cid:14) is, of course, negative.
6This argument seems to be supported in Arora, Bohn & Zhu (2005), who study the surprise
e⁄ect of distress announcements. Conditional on market information, they (cid:133)nd only 11 percent
of the distress (cid:133)rms(cid:146)equities and 18 percent of the distressed bonds to respond signi(cid:133)cantly.
The vast majority of prices are found to re(cid:135)ect the credit deterioration well before the distress
announcement.
8
2.3 Trading returns
The calculation of trading returns is fundamental to analyze how the risk and
return di⁄er across model assumptions and calibration methods. Since the CDS
position has a zero market value at initiation, trading returns must be calculated
by assuming that the arbitrageur has a certain level of initial capital. This as-
sumption allows us to hold (cid:133)xed the e⁄ects of leverage on the analysis. The initial
capital is used to (cid:133)nance the equity hedge, and credited or deducted as a result of
intermediate payments such as dividends or CDS premia. Each trade is equipped
with this initial capital and a limited liability assumption to ensure well-de(cid:133)ned
returns. Hence, each trade can be thought of as an individual hedge fund subject
to a forced liquidation when the total value of the portfolio becomes zero.7
Through the holding period the value of the equity position is straightforward,
but the value of the CDS position has to be calculated using equation (7). Since
secondarymarkettradingisverylimitedintheCDSmarketandnotcoveredbyour
dataset, we adopt the same simplifying assumption as Yu (2006), and approximate
c(t;T) with c(t;t+T). That is, we approximate a CDS contract maturing in four
years and ten months, say, with a freshly issued (cid:133)ve year spread. This should not
pose a problem since the di⁄erence between to points on the curve is likely to be
much smaller than the time-variation in spreads.
Yu (2006) (cid:133)nds his results insensitive to the exact size of transaction costs for
trading CDSs. We adopt his base case, and assume a 5 percent proportional bid-
ask spread on the CDS spread. The CDS market is likely to be the largest single
source of transaction costs for the arbitrageur. We therefore ignore transaction
costs on equities, which is reasonable under the static hedging scheme.
3 Data
Data on CDS spreads is provided by the ValuSpread database from Lombard Risk
Systems, dating back to July 1999. This data is also used by Lando & Mortensen
(2005) and Berndt, Jarrow & Kang (2006). The data consists of mid-market CDS
quotes on both sovereigns and corporates, with varying maturity, restructuring
clause, seniority and currency. For a given date and reference (cid:133)rm, the database
reportsacompositeCDSquotetogetherwithastandarddeviation. Thiscomposite
quote is calculated as a mid-market quote by obtaining quotes from up to 25
7This is reminiscent of potential large losses when marked to market, triggering margin calls
and forcing an early liquidation of positions.
9